Now, in general, \({\rm Tr}[\rho _{\rm cl}^2] \le 1\) , which considers the case of
any arbitrary classical distribution, and follows from the Liouville theorem.
The issue has been investigated earlier on in the context of quantum-classical
correspondence for decoherence induced by nonlinear couplings in bipartite systems
[1]}, [2]}, [3]}.
The elementary tool to explore this correspondence is the computation of entropies associated
with one of the two subsystems, such as the linear and the von Neumann entropies.
These entropies can be understood [4]} as a measurement of entanglement
produced over time by quantifying the degree of mixedness of one of the parties.
Note that, as it was pointed out by Schrödinger [5]}, as soon
as two systems start interacting, their state can no longer be described in a separate manner regardless of whether they are spatially separated (typical EPR condition
[6]}) or not.
An increase in the amount of entanglement means an increase of our lack of knowledge on the
system state, which, in turn, translates into an increase of the corresponding entropy measure.
This issue presents a remarkable similarity with former studies relating chaos with energy
relaxation rates
[7]}, [8]}, [9]}
or decoherence
[10]}, [11]}, [12]}, [13]}.
Furthermore, it is significant that entanglement dynamics of initially Gaussian states
can be fully described by means of a classical entropy in the classical limit [14]}.
